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J. M. CLARK &-0. A. L. TOTTEN. h GRADUATING ARCS UPON MATHEMATICAL AND OTHER INSTRUMENTS.,

No. 329,139. .Pa'tente-d on. 27, 1885.

N. PETERS Pnowuum m m. Washingmn. n, c.

NlTE STATES PATENT Prion.

JACOB M. CLARK, OF ELIZABETH, NEW JERSEY, AND CHARLES A.- L.

TOTTEN, OF UNITED STATES ARMY.

GRADUATING ARCS UPON MATHEMATlCAL AND OTHER INSTRUMENTS.

SPECIFICATIUN forming part of Letters Patent No. 329,139, dated October 27, 1885.

Application filed March 7, 1881.

To all whom it may concern.-

Be it known that we, JACOB M. CLARK, of Elizabeth, New Jersey, and CHARLES A. L. TOTTEN, of the United States army, have discovered and invented certain new and useful Improvements in Graduating Arcs upon Mathematical and other Instruments; and we do hereby declare the following to be a full, clear, and exact description of the same.

' This invention is particularly concerned in the discovery of the proper aliquot subdivision of the entire circumference and its application to the various mathematical and mechanical devices upon which graduated arcs are formed and employed.

The adoption of our system will change all the numerical values of arcs and functions which depend upon the three-hundred-and-sixty-degree subdivision in such away as to render them all new amounts. The change, however, can be easily effected, and depends upon a constant factor representing the ratio of our new subdivision to the old. Such a change is of course radical from a scientific standpoint, and looks toward harmonizing geometrical method and its numericalinterpretation and expression. These things are not now at unity. For instance, it is impossible to construct geometrically a nonagon, because the process would involve a trisection. This polygon is irrational, and cannot be drawn. Nevertheless, as 360Z-9::40 gives us a perfect numerical expression for the oneninth of a circle, we are apt to lose sight of geometric facts of vast importance while thus misinterpreted. Nothing can be gained in the long run by employing false methods, and

' it is far better to have geometry and its numerical expression go hand inhand than have them, as they are at present, radically at variance. Instead of dividing the circle into three hundred and sixty-degrees, therefore, we intend to divide it into two hundred and forty degrees, and in the case of the nonagon above noted it is seen that 240+9=26%, or =26.6666+ that is, it is irrational numerically as well as geometrically; for in one 3 form (twenty-six and two-thirds degrees) it involves trisection, and shows it by the denominator of its fractional quantity, and in the other (26.6666+) its repetend nature in- Serial No. 123,440. (No model.)

dicates that it can never be more than an ap proximation.

The adoption of our subdivision of the circle by practical mathematicians cannot possibly involve them in inconvenience, because the use of circular function, angle, 8m, in mathematical work is almost always intermediate-a means toward an end.

I11 the processes of surveying, drafting, astronomy, &c., it is the geometric angle and its function that the workman must employ if he expects to attain accuracy. This angle, we claim, is truthfully expressed by our system, and not by the one in use heretoforethat is, it is based upon what to the instrumerit-maker will be a practical problem. He certainly can subdivide a circumference into a rational number of parts with lessdanger of erring within a given degree than he could hope to do into an irrational number. So, too, numerically our system is truthful. Its interpretations of angle, function, and are are harmonious to their geometrical ideas, and this is not and cannot be the case of asys tern founded upon an irrational polygon and interpreted in numbers which necessarily in troduce a mental displacement.

We will now describe our discovery, after which we will be better able to point out more clearly how much and how little we differ from the present system.

Demonstration: By a strict geometrical pro cess it'is possible to obtain onerfifteenth of a circumference-that is, by comparing the onesixth with the one-tenth. Each of these arcs is thus rational. So, too, by as strict a process we can obtain one'sixteenth of a circumference (that is, by bisecting the one-eighth, which latter are is a direct function of the radius, its tangent being 1). .N ow, by examination it will be seen that the least common denominator of these resultant and rational arcs and angles is two hundred and forty, and that by their comparison we obtain the fractional unit i itself. Thus gglfizfi Now, in determining this subdivision of the circle we have exhausted all graphic (2'. 6., true geometric) methods without giving undue prominence to any one. The one-sixth is a natural division of the circle, the one-tenth comes from extreme and mean (ashas been shown in application for Letters 5' Patent numbered 119,5l4,"Fig. 7) from purely numerical relations, and the final one-twohundredand-fortieth has been obtained by comparison. This last angle is the resultant,

as it were, of all the above processes. It is 10 thus a most powerful circular denominator, as well as the greatest common divisor of the several subordinate and representative angles, both in their numeric and geometric relations. A regular polygon of two hundred and forty sides is thus not only a rational one, but one of most potent significance to the general mathematician. That we can attain this subdivision by purely geometrical means will now be illustrated by the following drawings, in which Figure 1 shows the several processes by means of which the one-two-hundred-and-fortieth of a circle is ultimately arrived at. Fig.

2 shows a circle subdivided into two hundred and forty parts, with a vernier for further subdivision; and Fig. 3 shows a section of a circle (three degrees) on such a subdivision, with the degrees further subdivided into ten minutes each.

Similar letters refer to similar parts throughout the several views.

Returning, now, to the consideration of Fig. 1, the following will be seen from simple in spection to be geometrically true:

First. As the radius H G is perpendicular to the diameter A E, H G E'is a quarter of a circle as measured by the arc H E.

Second. As the tangent I E is equal to the radius H G, the angle I G E is an eighth of a circle as measured by the arc J E.

Third. From J and E, with equal radii, describe the arcs intersecting at K. Through K draw K G. This angle will be a sixteenth of the circle as measured by the are L E.

45 Fourth. Lay off A S equal to the radius; then will S G A be a sixth ofthe circle. Lay off G P equal to one-half R; join P A; then, with P as a center and P A as a radius, de scribe the arc A Q, intersecting G H in Q. Q G will then be the chord of a tenth of the circle. Lay it off from S to U- Then will A U be the diiference between the one-sixthand' the one-tenth,or be equal to the fifteenth. Lay ofi E 0 equal to A U, then will 0 L be equal to the difference between E O 2 and (E L) gr, or E 0 will be the arc sought-that is, the two-hundredand-fortieth of the circleand the chord of it will be the side of a regular and rational polygon.

,iFifth. The fifteenth can be obtained geometrically and numerically in another way, as follows: Produce A E to, B, so that E B is equal to one-half R. Draw A- D: t R perpendicular to A E. Join D and B R, and through D draw D 0 parallel and equal to A B, and intersecting the circle at N. The distance N M,is the fifteenth of the circle, and may be compared withthe sixteenth, as above, and produce the twohundred-and-fortieth.

Sixth. Lay off the are 0 L two hundred and forty times upon the circle by its equal chords with a pair of dividers, and the pri-' mary subdivision of the circle into two hundred and forty parts is accomplished.

Seventh. By successive bisections of this arc we may obtain a primary subdivision into four hundred andeighty, nine hundred and sixty, one thousand nine hundred and twenty, or three thousand eight hundred and forty,&c., parts; or by deceminating each of the two hundred and forty parts the circle will be subdivided into two thousand four hundred, twenty-four thousand, &c., subordinate parts.

Eighth. By laying off the are 0 L ten times a twenty-fourth of the circle or a sector will result which will correspond with the hourly subdivisionsof the diurnal circle upon a twenty-four-hour clock.

Fig. 2 shows a circle subdivided, as above, into two hundred and forty parts or degrees, the quadrant H E alone having the degrees actually shown, and the rest of the subdivision being indicated.

nation of this subdivison of the circle with l collimators, telescopes, levels, magnetic needles, and other styles of indioes, for it would require an endless series of illustrations. I

1t is sufficient for our purpose to state here that we intend to graduate circles-such as are found upon any class of mathematical or other instruments-or parts of such circles, on the principle above described, and that it is the subdivision and its application to such instruments which we claim as new, and not the circle, nor the other features of instruments which are now in or mayhereafter comeinto use. 7 On the other hand, a regular polygon ofthree hundred and sixty sides is eternally irrational. method for obtaining a strict three-hundredand-sixtieth of a circle. It involves the trisection of an angle, and we shall square the circle itself as readily as accomplish it. It is true that all the arcs and circles which we now employ, or that the world hasemployed for full four thousand years, (excepting, perhaps, in compass-graduation,) are and have been based upon this three-hundred-and-sixtydegree subdivision. does notmake it a correct method. It is probable that for arithmetical reasons only the number three hundred and sixty seemed de- No geometer can demonstrate a Long usage, however,-

has been shifted from the geometer and asl ation and the processes of calculation. The

tronomer to the practical mechanic and instrument-maker, who has had to realize the approximation as best he could. The chronological system ofthe Babylonians, also based upon a three-hundred-and-sixty-day subdivision of the year, probably had its weight in determining the general use of this divisor. However, it is an erroneous subdivision, and this can be demonstrated as follows: Upon such a scale of division 5=, and 6= Therefore 6-5=l= Now, therefore, if we can construct the one-sixtieth and the one seventy-second of a circle, we can by comparison obtain the one three-hundredand-sixtieth. But while by two bisections from the one-fifteenth (which we know is rational) we can obtain a legitimate one-sixtieth, we cannot in any Way obtain the one seventy-second. This fractional part of the circle depends upon the one-ninth, and is the one-eighth thereof, of hence could be obtained by three successive bisections of the one-ninth, were not that angle itself (oneninth) an irrational one; but even were the three-hundred-andsixty-degree division a rational one we should still maintain the superiority of that of two hundred and forty degrees, because of its least common multiple and greatest common divisor properties, above noted. \Ve propose, therefore, to graduate circles into two hundred and forty parts, and to call each of these parts a degree, and we intend to place such graduations upon all mathematical instruments whatsoever. We intend even to box the mariners compass in this way, as well as the surveyors compass, to use it upon theodolites, transits, protractors, and, in fact, wherever now the three hundred-and sixty-degree subdivision is and has hitherto been employed. It is the only rational circummetric subdivision, and its introduction upon our instruments will result in vast simplification of mathematical processes, and be of other manifest practical advantages. Of course to facilitate its introduction we may have to re-edit in time all tables of functions, &c., now in use; but the simple numerical relation of our subdivision,two hundred and forty degrees, to the present one, three hundred and sixty degrees-to wit, two-thirds-will enable all the present tables to be used as Well with our instruments as with those for which the tables themselves were actually calculated. It is manifest that the simple introduction of the proper logarithm dependent upon two-thirds, and the constitution of the tables in hand, will render such tables immediately suitable for our instruments. For the further subdivision of the two hundredand forty degrees into subordinate parts we shall employ either simple bisection, or, preferably, the decimal Vernier. The latter will afford us manifest advantages in the construction of tables of functions and their use, and will be very Valuable in numerfollowing table, covering the latter method, seems best to realize the wants of modern science, and is entirely our own:

Rational Circummetric Graduation.

Name Cir. Sec. Deg. Min. Sec. 3ds. 4ths, &c. Symbol (B V 0 I 1/ 1 iv Value 1:24:240=2400=24000=240000=2400000, 8w.

1= 10: 100: 1000: 10000: 100000, &c.

1= 10: 100: 1000: 10000, &c. 1: 10: 100: 1000,&c.

1: 10,&c.

1,&c.

or omitting alternate terms- 1=245=24060=2400h00=1 1: 100: 10000:&e. 1: 100=&c.

1:8!- Referring, now, to the first table, it will be immediately seen that the system affords a most natural and decimal mode of writing, enumerating, and calculating. Take, for instance, the daily angular distance gained by the earth in its annual revolution about the sun. This daily angle 0: is expressed as follows: o =.657l255279+ degrees; but it is manifest that since a strict decimal system pervades the table the angle may also be readiv v vi vii viii ix x 5. 7. 1. 2. 5. 5. 2. 7; 9.+&c.;

X: 0: 6, 571, 255, 279, &c. The process of calculation is likewise similar.

For instance, the angular progress of the earth in thirty days 30 or is as follows:

1/ HI iv v vi vii viii in X 19. 7.

&c. The subdivision proposed being decimal below the primary division, the system, in fact,

presents all the facilities possessed by a pure decimal one-such as our monetary of dollars, dimes, cents, mills, &c.-while for special subdivision it admits of a duodecimal method, octenary, hexagonal, pentagonal, &c.

Having now described our invention, what we claim as new, and desire to secure by Letters Patent, is-

l. A scale for circular measure, the entire length of whose circumference is divided 'into two-hundred-andJorty-degree equal parts, as and for the purposes specified.

2. A scale for circular measure, the entire length of whose circumference is divided into two hundred and forty-degree equal parts, and each of these latter ones into other convenient aliquot parts, (decimal or otherwise,) as and for the purposes specified.

3. A circular scale for draftsmens use, divided into two hundred and forty equal parts, and suitable aliquot subdivisions, and ments, divided as described, and adapted for 10 bearing lines adapted to lay off the sides of application to mathematical and draftsmens rational and irrational polygons,or the chords instruments, as and for the purpose set forth. of their angles at the center, and vice versa, 7 5 as speclfied. X E- 4. A protractor consisting of a part of a v circular scale or are proportionally divided, Witnesses: as above set forth. LOYALL FARRAGUT.

5. A scale for circular or angular measure- J. T.- MAHON. 

